What is the range of the function $y=\sqrt{x+5}$ ?
The function given is \(y = \sqrt{x+5}\). To determine the range, we must identify all possible values that \(y\) can take. The expression under the square root, \(x+5\), must be non-negative for the function to be defined, meaning \(x + 5 \geq 0\), or \(x \geq -5\). This is the domain of the function.
The principal square root function \(\sqrt{u}\) is defined such that its output is always greater than or equal to 0 for any non-negative input \(u\). Since the smallest value that the expression \(x+5\) can take is 0 (occurring when \(x = -5\)), the smallest value the function \(y\) can produce is \(\sqrt{0} = 0\). As \(x\) increases towards infinity, \(x+5\) also increases towards infinity, and therefore \(\sqrt{x+5}\) increases towards infinity. Thus, the function takes all real values starting from 0.
Based on the analysis, the set of all possible output values (the range) is all real numbers \(y \geq 0\). Comparing this result with the given choices, we confirm that the correct option is \(y \geq 0\). The final answer is \(\boxed{y \geq 0}\).
In fields like meteorology, the square root function is often used to model relationships between variables, such as the relationship between wind speed and pressure. If you are calculating a physical property modeled by a square root function, understanding its range helps you determine the possible physical outcomes, such as identifying that a measurement (like speed or pressure) can never be negative. This ensures that your predictions align with the physical constraints of the real world.